Real class sizes

Abstract

In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group G have the same 2-part and the Sylow 2-subgroup of G satisfies certain condition, then G is solvable.

Finite groups whose real classes have prime-power size

“…Lemma 2.1. [12,Lemma 3.1] Let G be a finite group and suppose that ∆(G) has 2 connected components with vertex sets π 1 , π 2 , where…”

“…Proof. The first part is Theorem A and B of [12], the second is [4,Thorem 6.2] If all the real classes have prime-power size and V G = {q}, then structure of G is described in [10, Theorem C] if q = 2 and, if q > 2, by the following result. If |V G | > 1, the situation is more complex.…”

The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

“…For odd primes, the outcomes are not that satisfactory. Furthermore, it follows from ( [9], Theorem B) that, if the prime graph Γ r (G) of G is disconnected, then 2 must be a vertex of Γ r (G). These facts confirm repeatedly the importance and the special position of the prime 2 in the study of real classes in groups.…”